By (), we know the first term on the right hand side is a square integrable martingale.
The strong law of large number [1]} suggests that
\({\frac{1}{t}\sum _{i=0}^{t-1}({\bar{g}}_i{\bar{g}}_i^T) - \mathbb {E}[{\bar{g}}_i{\bar{g}}_i^T\mid \mathcal {F}_{i-1}]} = o{\sqrt{\frac{(\log t)^{1+\upsilon }}{t}}}.\) 
